Artificial Intelligence Modal Logic

Slides:



Advertisements
Similar presentations
Artificial Intelligence
Advertisements

Techniques for Proving the Completeness of a Proof System Hongseok Yang Seoul National University Cristiano Calcagno Imperial College.
Logic.
LDK R Logics for Data and Knowledge Representation Modal Logic Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia,
Knowledge Representation Methods
1 Conditional XPath, the first order complete XPath dialect Maarten Marx Presented by: Einav Bar-Ner.
Computability and Complexity 9-1 Computability and Complexity Andrei Bulatov Logic Reminder (Cnt’d)
CPSC 322, Lecture 20Slide 1 Propositional Definite Clause Logic: Syntax, Semantics and Bottom-up Proofs Computer Science cpsc322, Lecture 20 (Textbook.
1 CA 208 Logic Ex3 Define logical entailment  in terms of material implication  Define logical consequence |= (here the semantic consequence relation.
1 CA 208 Logic Logic Prof. Josef van Genabith Textbooks:  The Essence of Logic, John Kelly, Prentice Hall, 1997  Prolog Programming, Third Edition, Ivan.
Many Valued Logic (MVL) By: Shay Erov - 01/11/2007.
Artificial Intelligence 2004 Non-Classical Logics.
CS 4700: Foundations of Artificial Intelligence
EE1J2 – Discrete Maths Lecture 5 Analysis of arguments (continued) More example proofs Formalisation of arguments in natural language Proof by contradiction.
Relation, function 1 Mathematical logic Lesson 5 Relations, mappings, countable and uncountable sets.
Discrete Mathematics and Its Applications
LDK R Logics for Data and Knowledge Representation Context Logic Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia,
Theoretical basis of GUHA Definition 1. A (simplified) observational predicate language L n consists of (i) (unary) predicates P 1,…,P n, and an infinite.
Math 3121 Abstract Algebra I Section 0: Sets. The axiomatic approach to Mathematics The notion of definition - from the text: "It is impossible to define.
1st-order Predicate Logic (FOL)
Pattern-directed inference systems
Advanced Topics in Propositional Logic Chapter 17 Language, Proof and Logic.
Propositional Logic Dr. Rogelio Dávila Pérez Profesor-Investigador División de Posgrado Universidad Autónoma Guadalajara
0 What logic is or should be Propositions Boolean operations The language of classical propositional logic Interpretation and truth Validity (tautologicity)
CS344: Introduction to Artificial Intelligence Lecture: Herbrand’s Theorem Proving satisfiability of logic formulae using semantic trees (from Symbolic.
CS621: Artificial Intelligence Pushpak Bhattacharyya CSE Dept., IIT Bombay Lecture 28– Interpretation; Herbrand Interpertation 30 th Sept, 2010.
LDK R Logics for Data and Knowledge Representation Modal Logic Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia,
1 CA 208 Logic PQ PQPQPQPQPQPQPQPQ
Logical Agents Chapter 7. Outline Knowledge-based agents Logic in general Propositional (Boolean) logic Equivalence, validity, satisfiability.
CS6133 Software Specification and Verification
For Wednesday Read chapter 9, sections 1-3 Homework: –Chapter 7, exercises 8 and 9.
Artificial Intelligence “Introduction to Formal Logic” Jennifer J. Burg Department of Mathematics and Computer Science.
1 Finite Model Theory Lecture 1: Overview and Background.
Nikolaj Bjørner Microsoft Research DTU Winter course January 2 nd 2012 Organized by Flemming Nielson & Hanne Riis Nielson.
CS2351 Artificial Intelligence Bhaskar.V Class Notes on Knowledge Representation - Logical Agents.
Chapter 7. Propositional and Predicate Logic Fall 2013 Comp3710 Artificial Intelligence Computing Science Thompson Rivers University.
Computing & Information Sciences Kansas State University Lecture 12 of 42 CIS 530 / 730 Artificial Intelligence Lecture 12 of 42 William H. Hsu Department.
LDK R Logics for Data and Knowledge Representation Propositional Logic Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia,
Albert Gatt LIN3021 Formal Semantics Lecture 3. Aims This lecture is divided into two parts: 1. We make our first attempts at formalising the notion of.
ARTIFICIAL INTELLIGENCE [INTELLIGENT AGENTS PARADIGM] Professor Janis Grundspenkis Riga Technical University Faculty of Computer Science and Information.
Propositional Logic Rather than jumping right into FOL, we begin with propositional logic A logic involves: §Language (with a syntax) §Semantics §Proof.
ARTIFICIAL INTELLIGENCE Lecture 2 Propositional Calculus.
Propositional Logic Russell and Norvig: Chapter 6 Chapter 7, Sections 7.1—7.4.
Formal Semantics Purpose: formalize correct reasoning.
1 Lecture 3 The Languages of K, T, B and S4. 2 Last time we extended the language PC to the language S5 by adding two new symbols ‘□’ (for ‘It is necessary.
Lecture 041 Predicate Calculus Learning outcomes Students are able to: 1. Evaluate predicate 2. Translate predicate into human language and vice versa.
Metalogic Soundness and Completeness. Two Notions of Logical Consequence Validity: If the premises are true, then the conclusion must be true. Provability:
Propositional Logic Russell and Norvig: Chapter 6 Chapter 7, Sections 7.1—7.4 CS121 – Winter 2003.
Chapter Eight Predicate Logic Semantics. 1. Interpretations in Predicate Logic An argument is valid in predicate logic iff there is no valuation on which.
Logics for Data and Knowledge Representation ClassL (part 1): syntax and semantics.
Artificial Intelligence Logical Agents Chapter 7.
CS.462 Artificial Intelligence SOMCHAI THANGSATHITYANGKUL Lecture 04 : Logic.
Logical Agents. Outline Knowledge-based agents Logic in general - models and entailment Propositional (Boolean) logic Equivalence, validity, satisfiability.
CENG 424-Logic for CS Introduction Based on the Lecture Notes of Konstantin Korovin, Valentin Goranko, Russel and Norvig, and Michael Genesereth.
Propositional Logic (a.k.a. Sentential Logic)
Logics for Data and Knowledge Representation
Chapter 7. Propositional and Predicate Logic
Computer Science cpsc322, Lecture 20
ARTIFICIAL INTELLIGENCE
Semantics In propositional logic, we associate atoms with propositions about the world. We specify the semantics of our logic, giving it a “meaning”. Such.
Logics for Data and Knowledge Representation
Lesson 5 Relations, mappings, countable and uncountable sets
1st-order Predicate Logic (FOL)
Lesson 5 Relations, mappings, countable and uncountable sets
Logics for Data and Knowledge Representation
Chapter 7. Propositional and Predicate Logic
Computer Science cpsc322, Lecture 20
Bottom Up: Soundness and Completeness
Representations & Reasoning Systems (RRS) (2.2)
1st-order Predicate Logic (FOL)
Presentation transcript:

74.419 Artificial Intelligence Modal Logic see reference last slide

Syntax of Modal Logic (□ and ◊) Formulae in (propositional) Modal Logic ML: The Language of ML contains the Language of Propositional Calculus, i.e. if P is a formula in Propositional Calculus, then P is a formula in ML. If  and  are formulae in ML, then , , , , □, ◊ * are also formulae in ML. * Note: The operator ◊ is often later introduced and defined through □ .

Semantics of Modal Logic (□ and ◊) The semantics of a modal logic ML is defined through: a set of worlds W = {w1, w2, ..., wn}, an accessibility relation RWW, and an interpretation function : {0,1}

Semantics of Modal Logic ( and ) The interpretation in ML of a formula P, Q, ... of the propositional language of ML corresponds to its truth value in the "current world": w (P)=1 iff I(P) is true in w. w (PQ)=1 iff I(PQ) is true in w. ...

Semantics of Modal Logic (□ and ◊) We extend the semantics with an interpretation of the operators □ and ◊, specified relative to a "current world" w. For all wW: w (□)=1 iff w': (w,w')R  w' ()=1 ; 0 otherwise. w (◊)=1 iff w': (w,w')R  w' ()=1 ; Note: Often, the operator ◊ is defined in terms of □: ◊  □

Semantics of Modal Logic (□ and ◊) We can also prove the equivalence of □ and ◊ for our definitions above: w (□)=1 iff (w (□)=1) (or w (□)=0) iff w': (w,w')R  w' ()=1 iff w': (w,w')R  w' ()=0 iff w': (w,w')R  w' ()=1 iff w (◊)=1 This means: □  ◊ Exercise: Proof ◊  □ !

Semantics of Modal Logic (□ and ◊) Other logical operators are interpreted as usual, e.g. w (□)=1 iff w (□)=0

Semantics of ML - Complex Formulas The interpretation of a complex formula of ML is based on the interpretation of the atomic propositional symbols, and then composed using the interpretation function  defined above, e.g. w (□)=1 iff (w': (w,w')R  w' ()=1) iff w': (w,w')R  w' ()=0 Let's say   (PQ). w': (w,w')R  w' (PQ)=0 w': (w,w')R  (w' (P)=0  w' (Q)=0) "P or Q" is not necessarily true in world w, if there is a world w', accessible from w, in which P is false or Q is false.

Semantics of Modal Logic - Grounding The interpretation in ML of a formula P, Q, ... of the propositional language of ML corresponds to its truth value in the "current world": w (P)=1 iff I(P) is true in w. w (PQ)=1 iff I(PQ) is true in w. ...

Semantics of Modal Logic A formula  is satisfied in a world w of a Model M=<W,R,>, if it is true in this world wW under the given interpretation , i.e. w ()=1. M, w |=  A formula  is true in a Model M=<W,R,>, if it is satisfied in all worlds wW of M. M |=  A formula  is valid, if it is true in all Models. |=  A formula  is satisfiable, if it is satisfied in at least one world wW of one Model M=<W,R,>. (or: If its negation is not valid.)

Semantics of Modal Logic A formula  is satisfied in a world w of a Model M=<W,R,>, if it is true in this world under the given interpretation , i.e. w ()=1. M, w |=  A formula  is true in a Model M=<W,R,>, if it is satisfied in all worlds wW of M. M |=  A formula  is valid, if it is true in all Models.  |=  A formula  is satisfiable, if it is satisfied in at least one world wW of one Model M=<W,R,>. (or: If its negation is not valid.) A formula  is a consequence of a set of formulas  in M=<W,r,>, if in all worlds wW, in which  is satisfied,  is also satisfied.  |= 

Semantics of Modal Logic: Terminology Sometimes the term "frame" is used to refer to worlds and their connection through the accessibility relation: A frame <W, R> is a pair consisting of a non-empty set W (of worlds) and a binary relation R on W. A model <F, > consists of a frame F, and a valuation  that assigns truth values to each atomic sentence at each world in W.

Textbooks on (Modal) Logic Richard A. Frost, Introduction to Knowledge-Base Systems, Collins, 1986 (out of print) Comments: one of my favourite books; contains (almost) everything you need w.r.t. foundations of classical and non-classical logic; very compact, comprehensive and relatively easy to understand. Allan Ramsay, Formal Methods in Artificial Intelligence, Cambridge University Press, 1988 Comments: easy to read and to understand; deals also with other formal methods in AI than logic; unfortunately out of print; a copy is on course reserve in the Science Library.

Textbooks on (Modal) Logic Graham Priest, An Introduction to Non-Classical Logic, Cambridge University Press, 2001 Comments: the most poplar book (at least among philosophy students) on non-classical, in particular, (propositional) modal logic. Kenneth Konyndyk, Introductory Modal Logic, University of Notre-Dame Press, 1986 (with later re-prints) Comments: relatively easy and nice to read; contains propositional as well as first-order (quantified) modal logic, and nothing else.

Textbooks on (Modal) Logic J.C. Beall & Bas C. van Fraassen, Possibilities and Paradox, University of Notre-Dame Press, 1986 (with later re-prints) Comments: contains a lot of those weird things, you knew existed but you've never encountered in reality (during your university education). G.E. Hughes & M.J. Creswell, A New Introduction to Modal Logic, Routledge, 1996 Comments: Location: Elizabeth Dafoe Library, 2nd Floor, Call Number / Volume: BC 199 M6 H85 1996